Proof of Connected Diagrams

The logarithmic relationship is equivalent to $$Z[J]=\exp[iW[J]]$$ where $W$ is the sum of connected diagrams. This formula is trivial to prove via Taylor expansion of the exponential $$\exp(X) = \sum_{n=0}^\infty \frac{X^n}{n!} $$ If we substitute $i$ times the sum of all connected diagrams $iW$ for $X$ in this formula, the term $X^n/n!$ will simply produce the products of $n$ components, i.e. all disconnected (for $n\gt 2$) diagrams with $n$ components.

The combinatorial factor will work, too. Recall that when we evaluate Feynman diagrams, we have to divide by the symmetry factor. The symmetry group of a disconnected, $n$-component diagram includes the permutation group of all the $n$ components if the components are the same, that's why there is $1/n!$ in front of a "fixed single 1-component diagram" to the $n$-th power.

The extra symmetry group from permuting the components is reduced to the product of $n_i!$ over all subgroups of the group of $n$ components that contain the same diagram. But $$ \prod_i \frac{1}{n_i!} $$ is exactly what we get if we calculate $1/n!$ times the coefficient from the expansion of the $n$th-power of the sum of the connected diagrams.

Essentially, the answer to this is nothing but a consequence of the product and chain rule for (functional) derivatives. Think of W as some abstract object where all the Feynman diagrams (up to arbitrary orders) are stored. It does not matter what W looks like exactly. Acting on W[J] with functional derivatives with respect to J, you will start to generate some "unique" diagrams, in the sense that each sequence of applied derivatives with respect to J gives you another diagram, so for example \begin{equation} \frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \frac{\delta}{\delta J(x_3)} W[J] \end{equation} gives you the 3-point function with end-points $x_{1,2,3}$ etc. Now, think about taking some n functional derivatives from $e^{W[J]}$: \begin{equation} \frac{\delta}{\delta J(x_1)} \ldots \frac{\delta}{\delta J(x_n)} e^{W[J]} \end{equation} The first derivative gives you just $W[J] e^{W[J]}$. Now, you apply the second derivative, where you have to apply the product rule: \begin{equation} \frac{\delta}{\delta J(x_1)} \ldots \frac{\delta}{\delta J(x_{n-2})} \left[ \left( \frac{\delta}{\delta J(x_{n-1})} W[J] \right) + W[J] \frac{\delta}{\delta J(x_{n-1})} \right] e^{i W[z]} \end{equation} Now continue like that until you have taken all $n$ derivatives. You see that more and more terms arise. However, the only term that constantly gives you "new" diagrams is the one, where all derivatives are applied to $W[J]$ and none to the $e^{i W[J]}$ a second time. All the other diagrams are products of terms with fewer derivatives, applied on "different" $W[J]$ and hence correspond to a multiplication of several diagrams that are not full n-point functions.

So or instance, assume that after having applied the functional derivative some $k$-times straight on $W$ and then some $m := n-k$ times on another $e^{i W[J]}$ factor (just by doing product rule) you will get a product of a fully connected $k$-point function multiplied with a fully connected $m$-point function where $k+m = n$.

If you think for a second about permutations:

All permutations are a product of disjoint cycles. So you can write a permutation by multiplying cycles together, then dividing through by the number of ways you can stitch them together because that gives the same permutation.

So to get all permutations, you multiply cycles C by each other, which gives $C^n/n!$ and sum them up. In other words you can generate permutations by taking exponential of cycles, and so the log of permutations are cycles, the connected permutations.

This is made proper using combinatorial species and the symbolic method.

If you think about Z as a generating function for the combinatorial species of the feynman graphs, then taking a logarithm to get connected structures is exactly what you would expect. And if you look at the functional integration in a rough way, basically the same rules you learn about feynman diagrams are the ones given by manipulating the generators for these combinatorial species.